Solution Clustering in Random Satisfiability

For a large number of random constraint satisfaction problems, such as random k-SAT and random graph and hypergraph coloring, we have very good estimates of the largest constraint density for which solutions exist. All known polynomial-time algorithms for these problems, though, already fail to find solutions at much lower densities. To understand the origin of this gap we study how the structure of the space of solutions evolves in such problems as constraints are added. In particular, we show that for k ≥ 8, much before solutions disappear, they organize into an exponential number of clusters, each of which is relatively small and far apart from all other clusters. Moreover, inside each cluster most variables are frozen, i.e., take only one value. PACS. 02.50.-r Probability theory, stochastic processes, and statistics – 75.10.Nr Spin-glass and other random models